# Tangent Theorem 1

In this post, we discuss a theorem on tangent of circles.  We show that if a line is perpendicular to a radius of a circle at a point on the circle, then the line is tangent to the circle. In the figure below, line m is perpendicular to \$latex OA\$ at \$latex A\$. We show that it is a tangent to circle O.

Theorem

If a line is perpendicular to a radius of a circle at a point on the circle, then the line is tangent to the circle.

Given

Line \$latex m\$ is perpendicular to \$latex OA\$ at \$latex A\$.

Prove

Line \$latex m\$ is tangent to circle \$latex O\$

Proof

Let \$latex B\$ be another point on line \$latex m\$ other than \$latex A\$.

Since \$latex OA\$ is perpendicular \$latex m\$, triangle \$latex OAB\$ is a right triangle with hypotenuse \$latex OB\$. This means that \$latex OB > AB\$ and \$latex B\$ must be on the exterior of the circle. Therefore \$latex B\$ cannot lie on the circle and \$latex B\$ is the only point of m that is on the circle. So, \$latex m\$ is a tangent to the circle.

The converse of this statement is also true (see proof).